Density Matrix Renormalization Group Study of Random Dimerized Antiferromagnetic Heisenberg Chains

The effect of dimerization on the random antiferomagnetic Heisenberg chain with spin 1/2 is studied by the density matrix renormalization group method. The ground state energy, the energy gap distribution and the string order parameter are calculated. Using the finite size scaling analysis, the dimerization dependence of the these quantities are obtained. The ground state energy gain due to dimerization behaves as $u^a$ with $a > 2$ where $u$ denotes the degree of dimerization, suggesting the absence of spin-Peierls instability. It is explicitly shown that the string long range order survives even in the presence of randomness. The string order behaves as $u^{2\beta}$ with $\beta \sim 0.37$ in agreement with the recent prediction of real space renormalization group theory ($\beta =(3-\sqrt{5})/2 \simeq 0.382$). The physical picture of this behavior in this model is also discussed.Comment: 6 pages, 8 figures, to be published in Journal of the Physical Society of Japa

Random Magnetism in S=1/2S=1/2 Heisenberg Chains with Bond Alternation and Randomness on the Strong Bonds

The $S=1/2$ Heisenberg chains with bond alternation and randomness on the strong bonds are studied by the density matrix renormalization group method. It is assumed that the odd-th bond is antiferromagnetic with strength $J$ and even-th bond can take the values \JA and \JF (\JA > J > 0 > \JF) randomly. The ground state of this model interpolates between the Haldane and dimer phases via a randomness dominated intermediate phase. Based on the scaling of the low energy spectrum and mean field treatment of the interchain coupling, it is found that the magnetic long range order is induced by randomness in the intermediate regime. In the magnetization curves, there appears a plateau at the fractional value of the saturated magnetization. The fine structures of the magnetization curves and low energy spectrum are understood based on the cluster picture. The relation with the recent experiment for (CH$_3)_2$CHNH$_3$Cu(Cl$_x$Br$_{1-x})_3$ is discussed.Comment: 6 pages, 7 figures, Final version to appear in J. Phys. Soc. Jp

Density Matrix Renormalization Group Study of the S=1/2S=1/2 Antiferromagnetic Heisenberg Chains with Quasiperiodic Exchange Modulation

The low energy behavior of the $S=1/2$ antiferromagnetic Heisenberg chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. Based on the scaling behavior of the energy gap distribution, it is found that the ground state of this model belongs to the universality class different from that of the XY chain for which the precious mean exchange modulation is marginal. This result is consistent with the recent bosonization analysis of Vidal et al.Comment: 4 pages, 6 figure

Representations of the Double Burnside Algebra and Cohomology of the Extraspecial p-Group

Let E be the extraspecial p-group of order p^3 and exponent p where p is an odd prime. We determine the mod p cohomology of summands in the stable splitting of p-completed classifying space BE modulo nilpotence.Comment: 36 page

On Greenberg's LL-invariant of the symmetric sixth power of an ordinary cusp form

We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\geq4$, under some technical assumptions. This requires a "sufficiently rich" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to \GSp(4)_{/\QQ} of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.Comment: 20 pages, submitte